HopfionSeries Paper II - BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion

Paper I establishes numerically that the Euler–Lagrange minimum of the density-feedback energy Efb = Kfb · J4 satisfies Kfb/J4 = λ/21/3 to within O(h2) discretisation error (+0.020% on a 256×256 lattice), where λ = φ6. This paper addresses the statement analytically in three steps, introducing the generalised functional G[f; λeff] = Kfb[f] + λeff J4[f] and the ratio map H(λeff) := Kfb[fλeff]/J4[fλeff]. Existence For each λeff > 0, G has a critical point fλeff in the Q = 2 topological sector, via a mountain-pass argument in the energy space X2 (Theorem 2.1, subject to a concentration-compactness condition resolved in Paper III via Rellich–Kondrachov compactness). Monotonicity and uniqueness H is strictly…